Question

Let $p$ be an odd prime. Show that $2(p-3)!\bmod p=p-1$.

   Let $p$ be an odd prime. Show that $2(p-3)!\bmod p=p-1$.

Applied Algebra: Codes, Ciphers and Discrete Algorithms

Applied Algebra: Codes, Ciphers and Discrete Algorithms

Darel W. Hardy, Fred… 2nd Edition

Chapter 7, Problem 7

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This can be rewritten as \((p-1)! \equiv p-1 \pmod{p}\). Show more…

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Let $p$ be an odd prime. Show that $2(p-3)!\bmod p=p-1$.

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Key Concepts

Modular Arithmetic

Modular arithmetic is the system of arithmetic for integers, where numbers 'wrap around' upon reaching a certain value known as the modulus. In this system, two numbers are considered equivalent if they have the same remainder when divided by the modulus. It provides the framework for simplifying expressions and solving congruences, which is essential in analyzing problems involving remainders and residues in number theory.

Wilson's Theorem

Wilson's Theorem is a fundamental result in number theory which asserts that for any prime number p, the factorial (p-1)! is congruent to -1 modulo p. This theorem is not only a primality test but also a powerful tool for deriving congruences involving factorials. It is directly applicable to problems where factorial expressions need to be simplified modulo a prime.

Factorial Manipulation in Modular Context

Factorial manipulation in modular arithmetic involves breaking down factorial expressions into products of smaller terms to apply modular reductions and congruences effectively. This technique is particularly useful when combined with the properties of primes and the use of theorems like Wilson's Theorem. By decomposing a factorial into a product involving terms that are easier to manage modulo p, one can simplify and solve congruence equations that arise in number theoretic problems.

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